Editing Aleph number (section)

==Aleph-omega==
Aleph-omega is <math>\aleph_\omega = \sup\{\aleph_n| n \in \omega\} = \sup\{\aleph_n| n \in \{0, 1, 2,\cdots\}\}</math> where the smallest infinite ordinal is denoted as <math>\omega</math>. That is, the cardinal number <math>\aleph_\omega</math> is the [[least upper bound]] of <math>\sup\{\aleph_n| n \in \{0, 1, 2,\cdots\}\}</math>.

Notably, <math>\aleph_\omega</math> is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory ''not'' to be equal to the cardinality of the set of all [[real number]]s <math>2^{\aleph_0}</math>: For any natural number <math> n \ge 1 </math>, we can consistently assume that <math>2^{\aleph_0} = \aleph_n</math>, and moreover it is possible to assume that <math>2^{\aleph_0}</math> is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of <math>2^{\aleph_0}</math> is that it cannot equal certain special cardinals with [[cofinality]] <math>\aleph_0</math>. An uncountably infinite cardinal <math>\kappa</math> having cofinality <math>\aleph_0</math> means that there is a (countable-length) sequence <math>\kappa_0 \le \kappa_1 \le \kappa_2 \le \cdots</math> of cardinals <math>\kappa_i < \kappa</math> whose limit (i.e. its least upper bound) is <math>\kappa</math> (see [[Easton's theorem]]). As per the definition above, <math>\aleph_\omega</math> is the limit of a countable-length sequence of smaller cardinals.